Theoretical Mathematics & Applications

On Complex Vectors in C3 with Real Valued Scalar Product

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• Abstract 

  We consider a space S of complex vectors in C³ with physically relevant constraints and the corresponding representation of the group SO(3,C) acting on S. The constraints are introduced to provide real-valued and hyperbolically calculated vector magnitudes. Additionally, in order to acquire the benefits that real numbers provide, we introduce a real-valued scalar product in S using scalar product definition with relaxed conditions. This, in turn, leads to consider a specific SO(3,C) representation and restricted SO(3,C) action on S in order to keep the scalar product invariant.

  Mathematics Subject Classification : 51F25, 20C35, 22E10, 46C50
  Keywords: orthogonal group; polar decomposition; hyperbolic rotation